Just adding an incredibly larger amount of gases (the atmosphere is 5.5 * 10^18 kg) in the air doesn't increase global air pressure, it will just make more gases leak into the outer atmosphere over time. I am not sure how long that would take though.

Anyway, as you said, any project that doesn't involve significantly scaling down anthropic GHG emissions and human settlement footprint will have significant effects. Planting trillions of trees will result in dead trees and environments destroyed. CCS at scale will involve ungodly amounts of energy + materials. Algae farming will alter the ocean's biodiversity already affected by warming + acidity + overfishing.

It's time to acknowledge that the Earth doesn't owe us a couple of large cars per household, exotic foods, meat BBQs and regular plane trips.